A Simulated Annealing Based Algorithm with Collision Free Region for the Irregular Shape Packing Problem Andr´e Kubagawa Sato *,1 Thiago de Castro Martins *,2 Marcos de Sales Guerra Tsuzuki *,3 * Escola Polit´ecnica da Universidade de S˜ ao Paulo, S˜ao Paulo, Brazil. Mechatronics and Mechanical Systems Engineering Department Computational Geometry Laboratory Abstract: This paper investigates the irregular shape packing problem. The proposed algorithm constructively creates layouts from an ordered list of items and a placement heuristic. A moveable item is exclusively placed on a collision free region vertex. The container has a fixed width, while its length can change so that all items are placed on it. The objective is to find a layout of the set of items that minimizes the length of the container. The proposed algorithm has two hierarchical levels: core level with a simulated annealing algorithm, and the external level controlling the container length. The simulated annealing algorithm controls the ordered list of items and the placement of the items. If a feasible layout is found, the external level decreases the container length and applies the simulated annealing again. Otherwise, the container length is increased and the simulated annealing is applied again. Computational comparisons on benchmark problems show that the proposed algorithm generated highly competitive solutions. Moreover, our algorithm updated some best known results. Copyright c 2011 IFAC. Keywords: Simulated Annealing, Placement Problems, Optimization, Probabilistic Heuristic. 1. INTRODUCTION Cutting and packing are classic problems of finding the most efficient layout for a set of input items out of a given container with the objective of minimizing the waste material. The applications of cutting and packing, e.g. clothing, shipbuilding, and furniture industry are numer- ous. W¨ascher et al. (2007) have developed a typology that classifies the nesting problems according to dimension, objective, assortment of item type, as well as the number and nature of the container where the items are to be allocated. In this paper, we consider the irregular shape packing problem, classified as a two-dimensional irregular open dimension problem (ODP). The problem is NP-complete (Fowler et al., 1981) and as a result solution methodologies predominantly utilize heuristics. Although there are many different solution approaches presented in the literature, there appears to be two key strategies for representing and searching the solution space. The first approach represents the solution as an ordered list of items and applies a placement rule to construct the solution (Babu and Babu, 2001; Gomes and Oliveira, 2002; Burke et al., 2007, 2006). The second approach represents the solution as a physical layout on the container and moves items within the layout (Bennell 1 e-mail: andre.kubagawa@gmail.com. This author was partially supported by CNPq. 2 e-mail: thiago@usp.br. This author was partially supported by FAPESP (grant 2009/14699-0). 3 e-mail: mtsuzuki@usp.br. This author was partially supported by CNPq (grant 304258/2007-5) and FAPESP (grant 2009/07173-2). and Dowsland, 1999; Gomes and Oliveira, 2006; Egeblad et al., 2007). The former strategy is dependent on two critical characteristics of the algorithm; the placement rule and the placement order of the items. When irregular shaped items are involved an extra di- mension of complexity is generated by the geometry. The no-fit polygon is the principal approach for handling the geometry in cutting and packing problems with irregular shaped items (Burke et al., 2007; Bennell and Song, 2008; Costa et al., 2009). The no-fit polygon is used to ensure feasible layouts; i.e. layouts where the items do not overlap and fit inside the container. In this paper, the solution is represented as an ordered list of items and a constructive heuristic is applied se- quentially. A simulated annealing algorithm controls how the items are ordered in the list and where the items are constructively placed. The proposed algorithm has two hierarchical levels. The inner level solves the placement problem where the container has fixed dimensions. The external level reduces or increases the container dimen- sion that must be minimized. Experiments show that the proposed method is very efficient compared with other published methods. 2. PREVIOUS WORKS The no-fit polygon alone has limited utility in cutting and packing problems. Some recent works are using the collision free region to determine feasible layouts in con- tainers with fixed dimensions. Martins and Tsuzuki (2006) proposed a constructive heuristic where the collision free Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011 Copyright by the International Federation of Automatic Control (IFAC) 3968